576CHAPTER 14CALCULUS OF VECTOR-VALUED FUNCTIONS(ET CHAPTER 13)4π2R2v2=42R3GM1v2=R⇒M=Rv2G13. Mass of the Milky WayThe sun revolves around the center of mass of the Milky Way galaxy in an orbit thatis approximately circular, of radiusa≈2.8×1017km and velocityv≈250 km/s. Use the result of Exercise 12 toestimate the mass of the portion of the Milky Way inside the sun’s orbit (place all of this mass at the center of the orbit).SOLUTIONLetMbe the mass of the portion of the Milky Way inside the sun’s orbit, assuming that all this mass is atthe center of the sun’s orbit. By Exercise 12, the following equality holds:M=av2G.We substitute the valuesa=2.8×1020m,v=250×103m/sandG=6.673×10−11m3kg−1s−1and compute themassM.Thisgives:M=2.8·1020·(250·103)26.673·10−11=2.6225×1041kg.The mass of the sun is 1.989×1030kg, henceMis 1.32×1011times the mass of the sun (132 billions times the massof the sun).14. Conservation of EnergyThe total mechanical energy (kinetic plus potential) of a planet of massmorbiting a sunof massMwith positionrand speedv=kr0kisE=12mv2−GMmkrk11Use (2) and (9) to show thatEis conserved, that is,dEdt=0.We differentiate the total mechanical energyEwith respect to the timet, obtaining:=12m·2vdv−d±1krk¶=mkr0kdkr0k+1krk2dkrk=nmkr0kdkr0krk2dkrk(1)By formula (9),dkrk=r·r0krk. Substitution into (1) gives (fork=):=mkr0kdkr0krk2r·r0krk=mkr0kdkr0krk3r·r0=mkr0kdkr0±mkkrk3r¶·r0(2)By formula (2),r00=−kkrk3r, hence:=mkr0kdkr0k−mr00·r0(3)Using formula (9) forr0in place ofrgives:dkr0r0·r00kr0k(4)We substitute (4) into (3) to obtain:=mkr0kr0·r00kr0k−mr00·r0=mr0·r00−mr00·r0=015.Show that the total energy (11) of a planet in a circular orbit of radiusRisE/(2R).Hint:Use Exercise8.

This
preview
has intentionally blurred sections.
Sign up to view the full version.

SECTION14.6Planetary Motion According to Kepler and Newton(ET Section 13.6)577SOLUTIONThe total energy of a planet in a circular orbit of radiusRisE=12mv2−GMmkrk=12mv2−R(1)In Exercise 8 we showed thatv2=GMR(2)Substituting (2) in (1) we obtain:E=12mR−R=−12R2R.

This is the end of the preview.
Sign up
to
access the rest of the document.